Abstract
Infinitely many elliptic curves over \(\mathbf{Q }\) have a Galois-stable cyclic subgroup of order 4. Such subgroups come in pairs, which intersect in their subgroups of order 2. Let \(N_j(X)\) denote the number of elliptic curves over \(\mathbf{Q }\) with at least j pairs of Galois-stable cyclic subgroups of order 4, and height at most X. In this article we show that \(N_1(X) = c_{1,1}X^{1/3}+c_{1,2}X^{1/6}+O(X^{0.105})\). We also show, as \(X\rightarrow \infty \), that \(N_2(X)=c_{2,1}X^{1/6}+o(X^{1/12})\), the precise nature of the error term being related to the prime number theorem and the zeros of the Riemann zeta-function in the critical strip. Here, \(c_{1,1}= 0.95740\ldots \), \(c_{1,2}=- 0.87125\ldots \), and \(c_{2,1}= 0.035515\ldots \) are calculable constants. Lastly, we show no elliptic curve over Q has more than 2 pairs of Galois-stable cyclic subgroups of order 4.
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Authors' contributions
The authors are grateful to John Voight for several useful conversations, to Paul Pollack for his suggestion of using (3), and to two very thorough referees for their helpful corrections and suggestions. In particular, one referee suggested a better proof of Proposition 3 and the other a better proof of Lemma 4. The authors are also grateful to Mehdi Ahmadi for creating the figure. The authors made use of GP-PARI and Mathematica.
Funding
The first author is grateful for the hospitality of Santa Clara University and was funded by their Paul R. and Virginia P. Halmos Endowed Professorship in Mathematics and Computer Science.
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The first author is grateful for the hospitality of Santa Clara University and was funded by their Paul R. and Virginia P. Halmos Endowed Professorship in Mathematics and Computer Science
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Pomerance, C., Schaefer, E.F. Elliptic curves with Galois-stable cyclic subgroups of order 4. Res. number theory 7, 35 (2021). https://doi.org/10.1007/s40993-021-00259-9
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DOI: https://doi.org/10.1007/s40993-021-00259-9